System evolution of forward-backward multiplicity correlations in a multi-phase transport model

The initial geometry effect on forward-backward multiplicity correlations $C(N_{f},N_{b})$ is studied in relativistic collisions between light nuclei by using a multiphase transport model (AMPT). It is found that tetrahedron $^{16}$O + $^{16}$O gives a more uniform and symmetrical fireball which produces a more isotropic distribution of final particles after the expansion and evolution, and leads to a small $C(N_{f},N_{b})$. Forward-backward multiplicity correlation could be taken as a useful probe to distinguish the pattern of $\alpha$-clustered $^{16}$O in experiments by comparing the neighboring colliding nuclear systems like $^{14}$N + $^{14}$N and $^{19}$F + $^{19}$F.


I. INTRODUCTION
Relativistic heavy ion collisions produce an extreme hot and dense environment and provide a venue to understand the properties of early-stage quark matter in the Universe as well as the strong interaction [1][2][3][4][5][6]. Various probes such as hard probes, soft probes and electromagnetic probes etc [7][8][9][10][11][12] are proposed and measurements of heavy-ion collisions at Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) have put important constraints on the theoretical studies and model simulation. On the other hand, the αclustered structure of stable nuclei 12 C or 16 O has been one of the highly interesting topics in the heavy-ion community at lower energy [13][14][15][16][17][18][19][20], much progress has been achieved in recent years [21][22][23][24][25][26][27][28][29][30][31]. It is generally believed that the nuclear structure effect is significant only in low energy nuclear collisions. However, it was also proposed that this kind of nuclear structure phenomenon can be demonstrated in relativistic heavy-ion collisions through observables such as harmonic flows [21,22].
Two general classes of quantum chromodynamics processes combine to generate events in high energy hadron collisions. Multiple parton interactions picture the soft processes, together with the beam remnants, that are characterized by long range correlations. Short range correlation, which quickly diminishes as the pseudo-rapidity distance increases, manifests the hard processes that are arisen by the perturbatively described radiation or partonic scatterings and produced by single-and few-parton exchanges [32].
Due to the strong initial-state density fluctuations in the light nucleus, the space-time evolution of the produced matter in the final state fluctuates event to event.
These density fluctuations generate long-range correlations (LRC) at the early stages of the collision, well before the onset of any collective behavior, and appear as correlations of the multiplicity densities of produced particles separated in pseudo-rapidity (η) [33][34][35][36]. The forwardbackward (FB) correlation between final-state charged particle multiplicities in two separated η windows is a useful observable in high-energy hadron or nuclear collisions to study the dynamics of particle production mechanism [37][38][39][40] and may provide the information of collided nuclei if it is built up with exotic nuclear structure, such as α-cluster.
With that in mind, the forward-backward correlations are investigated through the AMPT model in different collision systems at center of mass energy √ s N N = 200 GeV and 6370 GeV. We find that system scan experiment could be a good way to distinguish the exotic α-clustered nuclear structure from the Woods-Saxon one. The rest of the paper is arranged as follows: In section II and section III a brief introduction to AMPT model and definition of forward-backward correlation is presented, respectively. Section IV presents the centrality, system as well as sphericity dependences of FB multiplicity correlation. More importantly, the section describes how to distinguish the structure of light nuclei through FB multiplicity correlation, and then a summary is given in Section V.

II. A BRIEF INTRODUCTION TO AMPT MODEL
A multi-phase transport model is developed to address the non-equilibrium many-body dynamics and aims at describing physics in relativistic heavy-ion collisions at arXiv:2110.04784v1 [hep-ph] 10 Oct 2021 RHIC [41,42]. It is also suitable to reproduce the results at LHC [43] including the pion HBT correlations [44], dihadron azimuthal correlations [45], collective flow [46][47][48], and strangeness production and correlation [49,50]. AMPT is a hybrid dynamic transport model, which consists of four main components: (a) the initial conditions including the spatial and momentum distributions of mini-jet partons and soft string excitation, which are obtained from the HIJING model; (b) partonic cascade [51], whereby interactions among partons are described by equations of motion for their Wigner distribution functions; (c) hadronization, which is conversion from the partonic to the hadronic matter; and (d) hadronic interactions, based on the ART (a relativistic transport) model [52], including baryon-baryon, baryon-meson, and meson-meson elastic and inelastic scatterings. Details of the AMPT model can be found in recent reviews [41,42].
The initial nucleon distribution in nuclei is configured in the HIJING model [53,54] with either a pattern of Woods-Saxon distribution or an exotic nucleon distribution which is embedded to study the α-clustered structure of 16 O. For details, a tetrahedral four-α clustering structure of 16 O is taken into account in this work and parameters of the tetrahedral structure are inherited from an extended quantum molecular dynamics (EQMD) model [15], which is extended from the quantum molecular dynamics (QMD) model. With the effective Pauli potential, EQMD model can give reasonable α-cluster configurations for 4N nuclei. For the four αs in the tetrahedral structure, we put them at the vertices with side length of 3.42 fm so that it gives a similar rms-radius (2.699 fm) to the Woods-Saxon configuration (2.726 fm) as well as the experimental data (2.6991 fm) [55], while nucleons inside each α are initialized by using the Woods-Saxon distribution introduced in the HIJING model.

III. FORWARD-BACKWARD MULTIPLICITY CORRELATION
One method to characterize the forward-backward multiplicity correlation can be described by the following definition [56,57] where N f and N b are the numbers of charged particles falling into the forward and backward pseudorapidity interval δη, respectively. The quantity C(N f , N b ) vanishes if there is no correlation between N f and N b , so that C(N f , N b ) measures the deviation from Poissonstatistical behavior. Of practical importance, the ratio is robust since it is independent of experimental efficiency as well as what fraction of particles are used.
Here two intervals separated symmetrically around η = 0 with pseudorapidity width δη are defined as "forward" (η > 0) and "backward" (η < 0). Correlations between multiplicities of charged particles are studied as a function of the gap between the windows η gap , i.e. the distance between lower and upper boundary of forward and backward η window.

IV. RESULTS AND DISCUSSION
A. System and Centrality dependence of FB correlation At each collision energy, C(N f , N b ) is found to decrease slowly with the increasing of η gap . It is found that the value of C(N f , N b ) increases with the collision energy. In p + p collision, the mean multiplicity in pseudorapidity window 0 < η < 0.8 is around 1 at √ s N N = 10 GeV, and it is around 3.5 at √ s N N = 5020 GeV. The quite large values of C(N f , N b ) in p + p collision systems show a strong correlation between forward bin and backward bin, which indicates that jet events could play an important role in C(N f , N b ). This argument also suggests that we should focus on the shape of final particle distribution in each event. Table I shows   both N f and N b are confined to a fairly small multiplicity range, which leads C(N f , N b ) to be very close to 0.
To illustrate the FB correlation above as a probe to distinguish the clustering light nuclei, we also compute the forward-backward multiplicity correlation C(N f , N b ) of different initial structures of 16 O as well as its neighbouring collisions systems, namely 14 N + 14 N and 19 F + 19 F. Figure 2 shows the FB multiplicity correlation coefficient C(N f , N b ) as a function of impact parameter b for different collision systems at √ s N N = 6370 GeV. In each collision system, C(N f , N b ) increases with the impact parameter b, which can be understood through that the relative fluctuation increases with b. It can be seen that the correlation functions C(N f , N b ) present different correlation strength in 16 O+ 16 O collisions with 16 O configuration either in Woods-Saxon distribution or α-clustered tetrahedron structure. To clearly investigate the effect from α-clustering structure, figure 3 shows C(N f , N b ) as a function of η gap with b < 2.0 fm for different collision systems at √ s N N = 6370 GeV. With the increasing system size, for light nuclei with Woods-Saxon distribution the C(N f , N b ) gradually decreases, close to zero, which can be understood that large systems tend to be binomial distribution, and the correlation between N f and N b becomes weaker.

B. Sphericity dependence of FB correlation
Considering that the shape of event has a great effect on the FB multiplicity correlation strength in p + p collision, to understand the source of multiplicity correlation in A+A collisions, it is necessary to perform a sphericity dependent analysis of C(N f , N b ) here. Transverse sphericity is defined as [60] S phero ⊥ ≡ π 2 4 min n=(nx,ny,0) where − → p ⊥ is the transverse momentum and n is the unit vector which minimizes S phero ⊥ . Numerically, the minimization is simplified by the observation that the n provides the minimal sum always coincides with the direction of one of the − → p ⊥ . Sphericity variable allows us to distinguish isotropic or spherical events which have high value of S phero ⊥ from the pencil shaped events which have low value of S phero ⊥ (eg. events are dominated with dijets).
Different from the sphericity distribution in p + p col-  N f , N b ). Table II shows the statistics of the number of partic- ipants corresponding to the systems in figure 3, N P part and N T part denotes the number of participants in projectile and target, respectively, σ(X) means the deviation of X, · · · represents the mean value. Since we are dealing with symmetric systems, only projectile side is shown for simplicity.
As we can see from the Table II, with increasing of system size, i.e. the Woods-Saxon configured systems from 14 N + 14 N, 16 O + 16 O, to 19 F + 19 F, the relative fluctuation of participant nucleon number becomes smaller. Fireball after collision is more uniform and symmetrical for the tetrahedral configuration 16 O in comparison with the Woods-Saxon one.
Assuming the distribution is binomial, its mean value should be proportional to system size O(N ), and its fluctuation is proportional to the O(N ), and then the relative fluctuation is proportional to 1/ O(N ). In this context, for the Woods-Saxon configured systems from 14 N + 14 N, 16 O + 16 O, to 19 F + 19 F, the relative fluctuation becomes smaller with the increasing of system size.
Observing that the particle number in a grand canonical ensemble in thermal equilibrium follows Poisson statistics, so C(N f , N b ) becomes smaller with the system size. Comparing the systems of tetrahedron and Woods-Saxon 16 O + 16 O, the mean values are similar but the distribution width (so the standard deviation of the number of final state charged particles) of tetrahedron configuration is narrower, which could explain the correlation coefficient C(N f , N b ) of 16 O with tetrahedral structure is slightly smaller than that with Woods-Saxon structure.

C. Eccentricity dependence of FB correlation
Considering that elliptic anisotropy of the collective component in both rapidity bins is highly correlated [62] and we are now discussing the relationship between initial configuration and C(N f , N b ), it is natural to check the initial anisotropic density characterized by complex eccentricity coefficients (ε n ) in the transverse plane. Figure 6 shows C(N f , N b ) as a function of ε 2 and ε 3 for different collision systems at √ s N N = 6370 GeV. ε n e inΦn = − r n e inφ / r n , where r = x 2 + y 2 and φ are the coordinate position and azimuthal angle of initial-state participant nucleons. Φ n is the initial participant plane angle. As we expected, from top to bottom are the 14 N + 14 N, 16 O + 16 O, and 19 F + 19 F systems with Woods-Saxon configuration. The larger system is easier to achieve thermal equilibrium, which gives a smaller C (N f , N b ). Focus on one system, the proportion of peripheral collision events in each bin is increasing from left to right, as we had expected more jet-like events in the final state gives larger C (N f , N b ).
From Table II we found that comparing with Woods-Saxon configuration, tetrahedron 16 O + 16 O gives a more uniform and symmetrical fireball, after the expansion and evolution it shows a more isotropic or spherical final particle distribution just as Figure 4 reveals. Events with spherical distribution give a small C (N f , N b ), whether in p + p collision or in 16  Based on the behavior of C(N f , N b ), we argue that the forward-backward multiplicity correlation measurement shall be a feasible observable to distinguish the αclustering structure through the systematic measurement for 16 O + 16 O collision and its neighbouring nuclear collisions.

V. SUMMARY
In summary, a systematic study on forward-backward multiplicity correlations C(N f , N b ) from large systems to small ones has been performed through the AMPT model. It is observed that the magnitude of FB correlation strength decreases from central to peripheral in Au + Au collisions at √ s N N = 200 GeV and the C( increases with the energy from √ s N N = 10 GeV, 200 GeV to 5.02 TeV is observed in p + p collision. Considering the shape of event has a great effect on the FB multiplicity correlation strength in p + p collision, a sphericity dependent study of C(N f , N b ) is performed in 16 O + 16 O system and C(N f , N b ) is found highly dependent on S phero ⊥ . When the events become more isotropic like, C(N f , N b ) becomes smaller, that's to say a uniform spread of final state particles gives a small C (N f , N b ).
Finally, the C(N f , N b ) of α-clustered 16 O + 16 O is compared with the results of Woods-Saxon type one, which shows visible difference between the two configurations. In the viewpoint of experimental measurements, we compare 16  Taken the initial participant nucleon into consideration, it seems the tetrahedral configuration 16 O is more uniform and symmetrical comparing to the Woods-Saxon one in the AMPT model. In one word, the FB correlation C(N f , N b ) could be proposed as a probe to distinguish the exotic α-clustering pattern experimentally and detailed measurement of pseudorapidity correlation in light nucleus collision also provides new constraints on the longitudinal dynamics of multiple parton interaction processes in the models. In light of the present study, we look forward to the related experiments being carried out in the future LHC experiments.